Notes on math, coding, and other stuff

I’ve been reading a lot of machine learning papers lately, and one thing I’ve noticed is that the vast majority of papers report positive results — “we used method X on problem Y, and beat the state-of-the-art results”. Very rarely do you see a paper that reports that something doesn’t work.

The result is publication bias — if we only publish the results of experiments that succeed, even statistically significant results could be due to random chance, rather than anything actually significant happening. Many areas of science are facing a replication crisis, where published research cannot be replicated.

There is some community discussion of encouraging more negative paper submissions, but as of now, negative results are rarely publishable. If you attempt an experiment but don’t get the results you expected, your best hope is to try a bunch of variations of the experiment until you get some positive result (perhaps on a special case of the problem), after which you pretend the failed experiments never happened. With few exceptions, any positive result is better than a negative result, like “we tried method X on problem Y, and it didn’t work”.

Why publication bias is not so bad

I just described a cynical view of academia, but actually, there’s a good reason why the community prefers positive results. Negative results are simply not very useful, and contribute very little to human knowledge.

Now why is that? When a new paper beats the state-of-the-art results on a popular benchmark, that’s definite proof that the method works. The converse is not true. If your model fails to produce good results, it could be due to a number of reasons:

Above: Only when everything is correct will you get positive results; many things can cause a model to fail. (Source)

So if you try method X on problem Y and it doesn’t work, you gain very little information. In particular, you haven’t proved that method X cannot work. Sure, you found that your specific setup didn’t work, but have you tried making modification Z? Negative results in machine learning are rare because you can’t possibly anticipate all possible variations of your method and convince people that all of them won’t work.

Searching for dragons

Suppose we’re scientists attending the International Conference of Flying Creatures (ICFC). Somebody mentioned it would be nice if we had dragons. Dragons are useful. You could do all sorts of cool stuff with a dragon, like ride it into battle.

“But wait!” you exclaim: “Dragons don’t exist!”

I glance at you questioningly: “How come? We haven’t found one yet, but we’ll probably find one soon.”

Your intuition tells you dragons shouldn’t exist, but you can’t articulate a convincing argument why. So you go home, and you and your team of grad students labor for a few years and publish a series of papers:

“We looked for dragons in China and we didn’t find any”

“We looked for dragons in Europe and we didn’t find any”

“We looked for dragons in North America and we didn’t find any”

…

Eventually, the community is satisfied that dragons probably don’t exist, for if they did, someone would have found one by now. But a few scientists still harbor the possibility that there may be dragons lying around in a remote jungle somewhere. We just don’t know for sure.

This remains the state of things for a few years until a colleague publishes a breakthrough result:

“Here’s a calculation that shows that any dragon with a wing span longer than 5 meters will collapse under its own weight”

You read the paper, and indeed, the logic is impeccable. This settles the matter once and for all: dragons don’t exist (or at least the large, flying sort of dragons).

When negative results are actually publishable

The research community dislikes negative results because they don’t prove a whole lot — you can have a lot of negative results and still not be sure that the task is impossible. In order for a negative result to be valuable, it needs to present a convincing argument why the task is impossible, and not just a list of experiments that you tried that failed.

This is difficult, but it can be done. Let me give an example from computational linguistics. Recurrent neural networks (RNNs) can, in theory, compute any function defined over a sequence. In practice, however, they had difficulty remembering long-term dependencies. Attempts to train RNNs using gradient descent ran into numerical difficulties known as the vanishing / exploding gradient problem.

Then, Bengio et al. (1994) formulated a mathematical model of an RNN as an iteratively applied function. Using ideas from dynamical systems theory, they showed that as the input sequence gets longer and longer, the result is more and more sensitive to noise. The details are technical, but the gist of it is that under some reasonable assumptions, training RNNs using gradient descent is impossible. This is a rare example of a negative result in machine learning — it’s an excellent paper and I’d recommend reading it.

Above: A Long Short Term Memory (LSTM) network handles long term dependencies by adding a memory cell (Source)

Soon after the vanishing gradient problem was understood, researchers invented the LSTM (Hochreiter and Schmidhuber, 1997). Since training RNNs with gradient descent was hopeless, they added a ‘latching’ mechanism that allows state to persist through many iterations, thus avoiding the vanishing gradient problem. Unlike plain RNNs, LSTMs can handle long term dependencies and can be trained with gradient descent; they are among the most ubiquitous deep learning architectures in NLP today.

After reading the breakthrough dragon paper, you pace around your office, thinking. Large, flying dragons can’t exist after all, as they would collapse under their own weight — but what about smaller, non-flying dragons? Maybe we’ve been looking for the wrong type of dragons all along? Armed with new knowledge, you embark on a new search…

XGBoost is a machine learning library that’s great for classification tasks. It’s often seen in Kaggle competitions, and usually beats other classifiers like logistic regression, random forests, SVMs, and shallow neural networks. One day, I was feeling slightly patriotic, and wondered: can XGBoost learn the Canadian flag?

Above: Our home and native land

Let’s find out!

Preparing the dataset

The task is to classify each pixel of the Canadian flag as either red or white, given limited data points. First, we read in the image with R and take the red channel:

Quick introduction to XGBoost

XGBoost implements gradient boosted decision trees, which were first proposed by Friedman in 1999.

Above: XGBoost learns an ensemble of short decision trees

The output of XGBoost is an ensemble of decision trees. Each individual tree by itself is not very powerful, containing only a few branches. But through gradient boosting, each subsequent tree tries to correct for the mistakes of all the trees before it, and makes the model better. After many iterations, we get a set of decision trees; the sum of the all their outputs is our final prediction.

In the first iteration, XGBoost immediately learns the two red bands at the sides:

After a few more iterations, the maple leaf starts to take form:

By iteration 60, it learns a pretty recognizable maple leaf. Note that the decision trees split on x or y coordinates, so XGBoost can’t learn diagonal decision boundaries, only approximate them with horizontal and vertical lines.

If we run it for too long, then it starts to overfit and capture the random noise in the training data. In practice, we would use cross validation to detect when this is happening. But why cross-validate when you can just eyeball it?

That was fun. If you liked this, check out this post which explores various classifiers using a flag of Australia.

The source code for this blog post is posted here. Feel free to experiment with it.

For the past few weeks, I’ve been working on the TensorFlow Speech Recognition Challenge on Kaggle. The task is to recognize a one-second audio clip, where the clip contains one of a small number of words, like “yes”, “no”, “stop”, “go”, “left”, and “right”.

In general, speech recognition is a difficult problem, but it’s much easier when the vocabulary is limited to a handful of words. We don’t need to use complicated language models to detect phonemes, and then string the phonemes into words, like Kaldi does for speech recognition. Instead, a convolutional neural network works quite well.

First Steps

The dataset consists of about 64000 audio files which have already been split into training / validation / testing sets. You are then asked to make predictions on about 150000 audio files for which the labels are unknown.

Actually, this dataset had already been published in academic literature, and people published code to solve the same problem. I started with GCommandPytorch by Yossi Adi, which implements a speech recognition CNN in Pytorch.

The first step that it does is convert the audio file into a spectrogram, which is an image representation of sound. This is easily done using LibRosa.

Above: Sample spectrograms of “yes” and “no”

Now we’ve converted the problem to an image classification problem, which is well studied. To an untrained human observer, all the spectrograms may look the same, but neural networks can learn things that humans can’t. Convolutional neural networks work very well for classifying images, for example VGG16:

Above: A Convolutional Neural Network (LeNet). VGG16 is similar, but has even more layers.

Voice Activity Detection

You might ask: if somebody already implemented this, then what’s there left to do other than run their code? Well, the test data contains “silence” samples, which contain background noise but no human speech. It also has words outside the set we care about, which we need to label as “unknown”. The Pytorch CNN produces about 95% validation accuracy by itself, but the accuracy is much lower when we add these two additional requirements.

For silence detection, I first tried the simplest thing I could think of: taking the maximum absolute value of the waveform and decide it’s “silence” if the value is below a threshold. When combined with VGG16, this gets accuracy 0.78 on the leaderboard. This is a crude metric because sufficiently loud noise would be considered speech.

Next, I tried running openSMILE, which I use in my research to extract various acoustic features from audio. It implements an LSTM for voice activity detection: every 0.05 seconds, it outputs a probability that someone is talking. Combining the openSMILE output with the VGG16 prediction gave a score of 0.81.

More improvements

I tried a bunch of things to improve my score:

Fiddled around with the neural network hyperparameters which boosted my score to 0.85. Each epoch took about 10 minutes on a GPU, and the whole model takes about 2 hours to train. Somehow, Adam didn’t produce good results, and SGD with momentum worked better.

Took 100% of the data for training and used the public LB for validation (don’t do this in real life lol). This improved my score to 0.86.

Trained an ensemble 3 versions of the same neural network with same hyperparameters but different randomly initialized weights and took a majority vote to do prediction. This improved the score to 0.87. I would’ve liked to train more, but other people in my research group needed to use the GPUs.

In the end, the top scoring model had a score of 0.91, which beat my model by 4 percentage points. Although not enough to win a Kaggle medal, my model was in the top 15% of all submissions. Not bad!

In many math competitions, all problems are weighted equally, even though the problems have very different difficulties. Sometimes, the harder problems are weighted more. But how should we assign weights to each problem?

Usually, the organizers make up weights based on how difficult they believe the problems are. However, they sometimes misjudge the difficulty of problems. Wouldn’t it be better if the weightings were determined from data?

Principal Component Analysis (PCA) is a statistical procedure that finds a transformation of the data that maximizes the variance. In our case, the first principal component gives a relative weighting of the problems that maximizes the variance of the total scores. This makes sense because we want to separate the good and bad students in a math contest.

IMO 2017 Data

The International Mathematics Olympiad (IMO) is an annual math competition for top high school students around the world. It consists of six problems, divided between two days: on each day, contestants are given 4.5 hours to solve three problems.

This year, 615 students wrote the IMO. Problems 1 and 4 were the easiest, with the majority of contestants receiving full scores. Problems 3 and 6 were the hardest: only 2 students solved the third problem. Problems 2 and 5 were somewhere in between.

This is a good dataset to play with, because the individual results show what each student scored for every problem.

Derivation of PCA for the 1-dimensional case

Let be a matrix containing all the data, where each column represents one problem. There are 615 contestants and 6 problems so has 615 rows and 6 columns.

We wish to find a weight vector such that the variance of is maximized. Of course, scaling up by a constant factor also increases the variance, so we need the constraint that .

First, PCA requires that we center so that the mean for each of the problems is 0, so we subtract each column by its mean. This transformation shifts the total score by a constant, and doesn’t affect the relative weights of the problems.

Now, is a vector containing the total scores of all the contestants; its variance is the sum of squares of its elements, or .

To maximize subject to , we take the singular value decomposition of . Then, the leftmost column of (corresponding to the largest singular value) gives that maximizes . This gives the first principal axis, and we are done.

Experiments

Running PCA on the IMO 2017 data produced interesting results. After re-scaling the weights so that the minimum possible score is 0 and the maximum possible score is 42 (to match IMO’s scoring), PCA recommends the following weights:

Problem 1: 9.15 points

Problem 2: 9.73 points

Problem 3: 0.15 points

Problem 4: 15.34 points

Problem 5: 5.59 points

Problem 6: 2.05 points

This is the weighting that produces the highest variance. That’s right, solving the hardest problem in the history of the IMO would get you a fraction of 1 point. P4 had the highest variance of the six problems, so PCA gave it the highest weight.

The scores and rankings produced by the PCA scheme are reasonably well-correlated with the original scores. Students that did well still did well, and students that did poorly still did poorly. The top students that solved the harder problems (2, 3, 5, 6) usually also solved the easier problems (1 and 2). The students that would be the unhappiest with this scheme are a small number of people who solved P3 or P6, but failed to solve P4.

Here’s a comparison of score distributions with the original and PCA scheme. There is a lot less separation between the best of the best students and the middle of the pack. It is easy to check that PCA does indeed produce higher variance than weighing all six problems equally.

Now, let me comment on the strange results.

It’s clearly absurd to give 0.15 points to the hardest problem on the IMO, and make P4, a much easier problem, be worth 100 times more. But it makes sense from PCA’s perspective. About 99% of the students scored zero on P3, so its variance is very low. Given that PCA has a limited amount of weight to “spend” to increase the total variance, it would be wasteful to use much of it on P3.

The PCA score distribution has less separation between the good students and the best students. However, by giving a lot of weight to P1 and P4, it clearly separates mediocre students that solve one problem from the ones who couldn’t solve anything at all.

In summary, scoring math contests using PCA doesn’t work very well. Although it maximizes overall variance, math contests are asymmetrical as we care about differentiating between the students on the top end of the spectrum.

This term, I was teaching a course on intro to theory of computation, and one of the topics was finite automatons — DFAs, NFAs, and the like. I began by writing down the definition of a DFA:

A deterministic finite automaton (DFA) is a 5-tuple where: is a finite set of states…

I could practically feel my students falling asleep in their seats. Inevitably, a student asked the one question you should never ask a theorist:

“So… how is this useful in real life?”

DFAs as a model of computation

I’ve done some theoretical research on formal language theory and DFAs, so my immediate response was why DFAs are important to theorists.

Above: A DFA requires O(1) memory, regardless of the length of the input.

You might have heard of Turing machines, which abstracts the idea of a “computer”. In a similar vein, regular languages describe what is possible to do with a computer with very little memory. No matter how long the input is, a DFA only keeps track of what state it’s currently in, so it only requires a constant amount of memory.

By studying properties of regular languages, we gain a better understanding of what is and what isn’t possible with computers with very little memory.

This explains why theorists care about regular languages — but what are some real world applications?

DFAs and regular expressions

Regular expressions are a useful tool that every programmer should know. If you wanted to check if a string is a valid email address, you might write something like:

/^([a-z0-9_\.-]+)@([\da-z\.-]+)\.([a-z\.]{2,6})$/

Behind the scenes, this regular expression gets converted into an NFA, which can be quickly evaluated to produce an answer.

You don’t need to understand the internals of this in order to use regular expressions, but it’s useful to know some theory so you understand its limitations. Some programmers may try to use regular expressions to parse HTML, but if you’ve seen the Pumping Lemma, you will understand why this is fundamentally impossible.

DFAs in compilers

In every programming language, the first step in the compiler or interpreter is the lexer. The lexer reads in a file of your favorite programming language, and produces a sequence of tokens. For example, if you have this line in C++:

The lexer uses a DFA to go through the source file, one character at a time, and emit tokens. If you ever design your own programming language, this will be one of the first things you will write.

Above: Lexer description for JSON numbers, like -3.05

DFAs for artificial intelligence

Another application of finite automata is programming simple agents to respond to inputs and produce actions in some way. You can write a full program, but a DFA is often enough to do the job. DFAs are also easier to reason about and easier to implement.

The AI for Pac-Man uses a four-state automaton:

Typically this type of automaton is called a Finite State Machine (FSM) rather than a DFA. The difference is that in a FSM, we do an action depending on the state, whereas in a DFA, we care about accepting or rejecting a string — but they’re the same concept.

DFAs in probability

What if we took a DFA, but instead of fixed transition rules, the transitions were probabilistic? This is called a Markov Chain!

Above: 3 state Markov chain to model the weather

Markov chains are frequently used in probability and statistics, and have lots of applications in finance and computer science. Google’s PageRank algorithm uses a giant Markov chain to determine the relative importance of web pages!

You can calculate things like the probability of being in a state after a certain number of time steps, or the expected number of steps to reach a certain state.

In summary, DFAs are powerful and flexible tools with myriad real-world applications. Research in formal language theory is valuable, as it helps us better understand DFAs and what they can do.

When you’re doing a statistical analysis, it’s easy to run into the multiple comparisons problem.

Imagine you’re analyzing a dataset. You perform a bunch of statistical tests, and one day you get a p-value of 0.02. This must be significant, right? Not so fast! If you tried a lot of tests, then you’ve fallen into the multiple comparisons fallacy — the more tests you do, the higher chance you get a p-value < 0.05 by pure chance.

They conducted 20 experiments and got a p-value < 0.05 on one of the experiments, thus concluding that green jelly beans cause acne. Later, other researchers will have trouble replicating their results — I wonder why?

What should they have done differently? Well, if they knew about the Bonferroni Correction, they would have divided the p-value 0.05 by the number of experiments, 20. Then, only a p-value smaller than 0.0025 is a truly significant correlation between jelly beans and acne.

Let’s dive in to explain why this division makes sense.

Šidák Correction

Time for some basic probability. What’s the chance that the scenario in the xkcd comic would happen? In other words, if we conduct 20 experiments, each with probability 0.05 of producing a significant p-value, then how likely will at least one of the experiments produce a significant p-value? Assume all the experiments are independent.

The probability of an experiment not being significant is , so the probability of all 20 experiments not being significant is . Therefore the probability of at least 1 of 20 experiments being significant is . Not too surprising now, isn’t it?

We want the probability of accidentally getting a significant p-value by chance to be 0.05, not 0.64 — the definition of p-value. So flip this around — we need to find an adjusted p-value to give an overall p-value 0.05:

Solving for :

Okay, this seems reasonably close to 0.0025. In general, if the overall p-value is and we are correcting for comparisons, then

This is known as the Šidák Correction in literature.

Bonferroni Correction

Šidák’s method works great, but eventually people started complaining that Šidák’s name had too many diacritics and looked for something simpler (also, it used to be difficult to compute nth roots back when they didn’t have computers). How can we approximate this formula?

Approximate? Use Taylor series, of course!

Assume is constant, and define:

We take the first two terms of the Taylor series of centered at 0:

Now and so . Therefore,

That’s the derivation for the Bonferroni Correction.

Since we only took the first two terms of the Taylor series, this produces a that’s slightly lower than necessary.

In the real world, is close to zero, so in practice it makes little difference whether we use the exact Šidák Correction or the Bonferroni approximation.

That’s it for now. Next time you do multiple comparisons, just remember to divide your p-value by . Now you know why.

This week I participated in the Porto Seguro Kaggle competition. Basically, you’re asked to predict a binary variable — whether or not an insurance claim will be filed — based on a bunch of numerical and categorical variables.

With over 5000 teams entering the competition, it was the largest Kaggle competition ever. I guess this is because it’s a fairly well-understood problem (binary classification) with a reasonably sized dataset, making it accessible to beginning data scientists.

Kaggle is a machine learning competition platform filled with thousands of smart data scientists, machine learning experts, and statistics PhDs, and I am not one of them. Still, I was curious to see how my relatively simple tools would fare against the sophisticated techniques on the leaderboard.

The first thing I tried was logistic regression. All you had to do was load the data into memory, invoke the glm() function in R, and output the predictions. Initially my logistic regression wasn’t working properly and I got a negative score. It took a day or so to figure out how to do logistic regression properly, which got me a score of 0.259 on the public leaderboard.

Next, I tried gradient boosted decision trees, which I had learned about in a stats class but never actually used before. In R, this is simple — I just needed to change the glm() call to gbm() and fit the model again. This improved my score to 0.265. It was near the end of the competition so I stopped here.

At this point, the top submission had a score of 0.291, and 0.288 was enough to get a gold medal. Yet despite being within 10% of the top submission in overall accuracy, I was still in the bottom half of the leaderboard, ranking in the 30th percentile.

The public leaderboard looked like this:

Above: Public leaderboard of the Porto Seguro Kaggle competition two days before the deadline. Line in green is my submission, scoring 0.265.

This graph illustrates the nature of this competition. At first, progress is easy, and pretty much anyone who submitted anything that was not “predict all zeros” got over 0.200. From there, you make steady, incremental progress until about 0.280 or so, but afterwards, any further improvements is limited.

The top of the leaderboard is very crowded, with over 1000 teams having the score of 0.287. Many teams used ensembles of XGBoost and LightGBM models with elaborate feature engineering. In the final battle for the private leaderboard, score differences of less than 0.001 translated to hundreds of places on the leaderboard and spelled the difference between victory and defeat.

Above: To run 90% as fast as Usain Bolt, you need to run 100 meters in 10.5 seconds. To get 90% of the winning score in Kaggle, you just need to call glm().

This pattern is common in Kaggle and machine learning — often, a simple model can do quite well, at least the same order of magnitude as a highly optimized solution. It’s quite remarkable that you can get a decent solution with a day or two of work, and then, 5000 smart people working for 2 months can only improve it by 10%. Perhaps this is obvious to someone doing machine learning long enough, but we should look back and consider how rare this is. The same does not apply to most activities. You cannot play piano for two days and become 90% as good as a concert pianist. Likewise, you cannot train for two days and run 90% as fast as Usain Bolt.

Simple models won’t win you Kaggle competitions, but we shouldn’t understate their effectiveness. Not only are they quick to develop, but they are also easier to interpret, and can be trained in a few seconds rather than hours. It’s comforting to see how far you can get with simple solutions — the gap between the best and the rest isn’t so big after all.